September 13, 2024

**Overview: ***Mastering quadratic equation questions is necessary to crack the exam with excellent scores. This article provides important quadratic equation questions for IPMAT 2025 to help you practice. Read on to enhance your problem-solving skills and ace the exam!*

In most entrance exams, you'll encounter a mathematics section that includes questions from various concepts, with the quadratic equation being one of the essential topics.

Although quadratic equations might seem complex at first glance, they can be solved quickly and efficiently using the correct formulas and methods.

This post will guide you through essential quadratic equation questions for the **IPMAT entrance exam**, providing several examples, solutions, and practice papers to help you master this topic.

Quadratic equations are a type of equation in algebra that can be rearranged in standard form as ax^{2}+bx+c=0 where x represents as unknown, and a, b, and c represent known numbers, and a ≠ 0.

If a = 0, the equation is linear, not quadratic, as there is no ax^2 term.

**Examples of the standard form of a quadratic equation (ax² + bx + c = 0) include:**

- 6x² + 11x - 35 = 0
- x² -x - 3 = 0
- 2x² - 4x - 2 = 0
- -4x² - 7x +12 = 0
- 5x² - 2x - 9 = 0
- 20x² -15x - 10 = 0

Here is the list of questions curated from **previous year's IPMAT Question Papers**.

The subject mentor from Supergrads has solved the questions below with a detailed explanation.

Solve these quadratic equations and enhance your preparation for the upcoming IPMAT exam.

Q1. If 𝛼 ≠ 𝛽 but α ^{2} = 5α − 3 and β ^{2} = 5β − 3 then the equation whose roots are 𝛼/𝛽 and 𝛽/𝛼 is

- (a) 3x
^{2}− 25x +3 = 0 - (b) x
^{2}+ 5𝑥 −3 = 0 - (c) x
^{2}− 5𝑥 +3 = 0 - (d) 3𝑥
^{2}− 19𝑥 + 3 = 0

**Answer: D**

Q2. Difference between the corresponding roots of x ^{2} + ax+ b = 0 and x ^{2} + bx + 𝑎 = 0 is same and 𝑎 ≠ 𝑏, then

- (a) 𝑎 + 𝑏 + 4 = 0
- (b) 𝑎 + 𝑏 − 4 = 0
- (c) 𝑎 − 𝑏 − 4 = 0
- (d) 𝑎 − 𝑏 + 4 = 0

**Answer: A**

Q3. If p and q are the roots of the equation x^{2} + px + q = 0 then

- (a) 𝑝 = 1, 𝑞 = −2
- (b) 𝑝 = 0, 𝑞 = 1
- (c) 𝑝 = −2, 𝑞 = 0
- (d) 𝑝 = −2, 𝑞 = 1

**Answer: A**

Q4. If a , b , c are distinct positive real numbers and a^{2} + b ^{2} + c ^{2} = 1 then 𝑎𝑏 + 𝑏𝑐 + 𝑐𝑎 is

- (a) less than 1
- (b) equal to 1
- (c) greater than 1
- (d) any real no

**Answer: A**

Q5. The value of a for which one root of the quadratic equation (a^{2} 2 − 5a+ 3)x ^{2}2 + (3a − 1)x + 2 = 0 is twice as large as the other is

- (a) -2/3
- (b) 1/3
- (c) -1/3
**(d) 2/3**

**Answer: D**

Q6. If the sum of the roots of the quadratic equation ax^{2} +bx + c = 0 is equal to the sum of the squares of their reciprocals, then a/c, b/a and c/b are in

- (a) geometric progression
- (b) harmonic progression
- (c) arithmetic-geometric progression
- (d) arithmetic progression

**Answer: B**

Q7. Let two numbers have an arithmetic mean nine and geometric mean 4 . Then these numbers are the roots of the quadratic equation

- (a) x
^{2}+ 18𝑥 −16 = 0 - (b) x
^{2}− 18𝑥 +16 = 0 - (c) x
^{2}+ 18𝑥 +16 = 0 - (d) x
^{2}− 18𝑥 −16 = 0

**Answer: B**

Q8. If (1 −𝑝) is a root of quadratic equation x^{2} + 𝑝𝑥 +(1 −𝑝) = 0 then its roots are

- (a) 0, -1
- (b) -1, 1
- (c) 0, 1
- (d) -1, 2

**Answer: D**

Q9. If one root of the equation x^{2}+ 𝑝𝑥 + 12 = 0 is 4 while the equation x ^{2} + 𝑝𝑥 + 𝑞 = 0 has equal roots, then the value of q is

- (a) 3
- (b) 12
- (c) 49/4
- (d) 4

**Answer: C**

Q10. If the roots of the equation x^{2} −𝑏𝑥 + 𝑐 = 0 be two consecutive integers, then b ^{2} −4𝑐 equals

- (a) 3
- (b) -2
- (c) 1
- (d) 2

**Answer: C**

Mastering quadratic equations is essential for cracking the IPMAT exam. Regular practice with various types of quadratic equation questions can significantly enhance your problem-solving skills and boost your confidence.

Solve the quadratic equation: x2−5x+6=0x^2 - 5x + 6 = 0x2−5x+6=0.

Solve the quadratic equation: 2x2+3x−2=02x^2 + 3x - 2 = 02x2+3x−2=0.

Find the roots of the quadratic equation: x2+4x+4=0x^2 + 4x + 4 = 0x2+4x+4=0.

Solve the quadratic equation: x2−2x−8=0x^2 - 2x - 8 = 0x2−2x−8=0.

Solve the quadratic equation: 3x2+7x+2=03x^2 + 7x + 2 = 03x2+7x+2=0.

For more quadraticeEquation questions, download the quadratic equation questions for IPMAT pdf, which includes questions and solutions.

Enhance your **preparation for IPMAT** by solving these Quadratic Equation Questions for IPMAT and score **good marks** in the mathematics section.

You can solve the below questions using various methods; one such is by factorization.

- Put all the terms on one side of the equal sign, leaving zero on the other side.
- Do the factorization.
- Set each factor equal to zero.
- Solve each of these equations.
- Check your solution by inserting your answer into the original equation.

Here is the list of formulas that you can use to solve **IPMAT** Questions.

- The standard form of a quadratic equation is ax
^{2}+ bx + c = 0 - The discriminant of the quadratic equation is D = b
^{2 }- 4ac - For D > 0 the roots are real and distinct.
- For D = 0 the roots are real and equal.
- For D < 0 the roots do not exist, or the roots are imaginary.
- The formula to find the roots of the quadratic equation is x =
- The sum of the roots of a quadratic equation is α + β = -b/a = - Coefficient of x/ Coefficient of x
^{2}. - The product of the Root of the quadratic equation is αβ = c/a = Constant term/ Coefficient of x
^{2} - The quadratic equation having roots α, β, is x
^{2}- (α + β)x + αβ = 0. - For positive values of a (a > 0), the quadratic expression f(x) = ax
^{2 }+ bx + c has a minimum value at x = -b/2a. - For negative value of a (a < 0), the quadratic expression f(x) = ax
^{2 }+ bx + c has a maximum value at x = -b/2a. - For a > 0, the range of the quadratic equation ax
^{2}+ bx + c = 0 is [b^{2}- 4ac/4a, ∞) - For a < 0, the range of the quadratic equation ax
^{2}+ bx + c = 0 is : (∞, -(b^{2}- 4ac)/4a]

Mastering quadratic equation questions is essential to excel in the IPMAT exam. Quadratic equations, though initially appearing complex, can be solved efficiently with the right approach. This section will guide you through an **effective preparation** strategy, incorporating important concepts, formulas, and problem-solving techniques to help you ace the quadratic equation questions in the IPMAT exam.

Before diving into practice, ensure you understand the fundamental concepts and formulas related to quadratic equations:

**1. Standard Form**: A quadratic equation is generally written as ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0, where xxx is the variable, and a,b,a, b,a,b, and ccc are constants with a≠0a \neq 0a=0.

**2. Discriminant**: The discriminant (DDD) of the quadratic equation ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 is given by D=b2−4acD = b^2 - 4acD=b2−4ac. The discriminant determines the nature of the roots:

- D>0D > 0D>0: Two distinct real roots
- D=0D = 0D=0: Two equal real roots
- D<0D < 0D<0: No real roots (roots are complex)

**3. Roots Formula**: The roots of the quadratic equation ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 can be found using the quadratic formula:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac

**4. Sum and Product of Roots**:

- Sum of the roots (α+β\alpha + \betaα+β): −ba-\frac{b}{a}−ab
- Product of the roots (αβ\alpha \betaαβ): ca\frac{c}{a}ac

To solve quadratic equation questions effectively, use these methods:

**Factorization**:

- Rewrite the equation in the form ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0.
- Find two numbers that
**multiply**to acacac and add up to bbb. - Factorize and solve for xxx.

**Completing the Square**:

- Rewrite the equation in the form ax2+bx=−cax^2 + bx = -cax2+bx=−c.
- Add and subtract (b2a)2\left( \frac{b}{2a} \right)^2(2ab)2 on both sides.
- Solve for xxx after simplifying.

**Using the Quadratic Formula**:

- Directly apply the quadratic formula to find the roots.

Consistent practice is crucial for mastering quadratic equations. Work on various types of questions, including those involving:

- Basic factorization
- Special cases with real, equal, and complex roots
- Application of the quadratic formula

Here are a few example problems to get you started:

**1. Solve**: x2−5x+6=0x^2 - 5x + 6 = 0x2−5x+6=0

- Factorize to (x−2)(x−3)=0(x - 2)(x - 3) = 0(x−2)(x−3)=0
- Roots: x=2x = 2x=2 and x=3x = 3x=3

**2. Solve**: 2x2+3x−2=02x^2 + 3x - 2 = 02x2+3x−2=0

- Using the quadratic formula: x=−3±9+164x = \frac{-3 \pm \sqrt{9 + 16}}{4}x=4−3±9+16
- Roots: x=12x = \frac{1}{2}x=21 and x=−2x = -2x=−2

**3. Solve**: x2+4x+4=0x^2 + 4x + 4 = 0x2+4x+4=0

- Completing the square: (x+2)2=0(x + 2)^2 = 0(x+2)2=0
- Root: x=−2x = -2x=−2

Practice solving quadratic equations within a time limit to improve speed and accuracy. Allocate specific times for each problem and gradually reduce the time as you become more proficient.

Incorporate quadratic equation questions into your mock tests to simulate exam conditions. Regular revision of key concepts and formulas will reinforce your understanding and boost your confidence.

Mastering quadratic equations is crucial for excelling in the IPMAT exam. By understanding fundamental concepts, practicing various problem-solving techniques, and regularly revising key formulas, you can significantly enhance your mathematical abilities.

- Essential Concepts: Understand the standard form of quadratic equations, the discriminant, and the quadratic formula to solve various types of problems.
- Problem-Solving Techniques: Use factorization, completing the square, and the quadratic formula to solve quadratic equations efficiently.
- Regular Practice: Consistent practice with diverse types of quadratic equation questions is crucial for improving problem-solving skills and speed.
- Time Management: Practice solving questions within a time limit to enhance speed and accuracy during the exam.
- Mock Tests and Revision: Incorporate quadratic equation questions in mock tests and regularly revise key concepts and formulas to reinforce understanding and boost confidence.

Frequently Asked Questions

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